Finite Math Examples

$x + 3 y - 3 z + 4 w = 3$ , $y + z - w = 4$ , $2 z + 2 w = 6$ , $2 w = 4$

Step 1

Divide each term in $2 w = 4$ by $2$ and simplify.

Tap for more steps...

Step 1.1

Divide each term in $2 w = 4$ by $2$ .

$\frac{2 w}{2} = \frac{4}{2}$

$x + 3 y - 3 z + 4 w = 3$

$y + z - w = 4$

$2 z + 2 w = 6$

Step 1.2

Simplify the left side.

Tap for more steps...

Step 1.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.1.1

Cancel the common factor.

$\frac{2 w}{2} = \frac{4}{2}$

$x + 3 y - 3 z + 4 w = 3$

$y + z - w = 4$

$2 z + 2 w = 6$

Step 1.2.1.2

Divide $w$ by $1$ .

$w = \frac{4}{2}$

$x + 3 y - 3 z + 4 w = 3$

$y + z - w = 4$

$2 z + 2 w = 6$

$w = \frac{4}{2}$

$x + 3 y - 3 z + 4 w = 3$

$y + z - w = 4$

$2 z + 2 w = 6$

$w = \frac{4}{2}$

$x + 3 y - 3 z + 4 w = 3$

$y + z - w = 4$

$2 z + 2 w = 6$

Step 1.3

Simplify the right side.

Tap for more steps...

Step 1.3.1

Divide $4$ by $2$ .

$w = 2$

$x + 3 y - 3 z + 4 w = 3$

$y + z - w = 4$

$2 z + 2 w = 6$

$w = 2$

$x + 3 y - 3 z + 4 w = 3$

$y + z - w = 4$

$2 z + 2 w = 6$

$w = 2$

$x + 3 y - 3 z + 4 w = 3$

$y + z - w = 4$

$2 z + 2 w = 6$

Step 2

Replace all occurrences of $w$ with $2$ in each equation.

Tap for more steps...

Step 2.1

Replace all occurrences of $w$ in $x + 3 y - 3 z + 4 w = 3$ with $2$ .

$x + 3 y - 3 z + 4 (2) = 3$

$w = 2$

$y + z - w = 4$

$2 z + 2 w = 6$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Multiply $4$ by $2$ .

$x + 3 y - 3 z + 8 = 3$

$w = 2$

$y + z - w = 4$

$2 z + 2 w = 6$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

$y + z - w = 4$

$2 z + 2 w = 6$

Step 2.3

Replace all occurrences of $w$ in $y + z - w = 4$ with $2$ .

$y + z - (2) = 4$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

$2 z + 2 w = 6$

Step 2.4

Simplify the left side.

Tap for more steps...

Step 2.4.1

Multiply $- 1$ by $2$ .

$y + z - 2 = 4$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

$2 z + 2 w = 6$

$y + z - 2 = 4$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

$2 z + 2 w = 6$

Step 2.5

Replace all occurrences of $w$ in $2 z + 2 w = 6$ with $2$ .

$2 z + 2 (2) = 6$

$y + z - 2 = 4$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

Step 2.6

Simplify the left side.

Tap for more steps...

Step 2.6.1

Multiply $2$ by $2$ .

$2 z + 4 = 6$

$y + z - 2 = 4$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

$2 z + 4 = 6$

$y + z - 2 = 4$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

$2 z + 4 = 6$

$y + z - 2 = 4$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

Step 3

Solve for $z$ in $2 z + 4 = 6$ .

Tap for more steps...

Step 3.1

Move all terms not containing $z$ to the right side of the equation.

Tap for more steps...

Step 3.1.1

Subtract $4$ from both sides of the equation.

$2 z = 6 - 4$

$y + z - 2 = 4$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

Step 3.1.2

Subtract $4$ from $6$ .

$2 z = 2$

$y + z - 2 = 4$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

$2 z = 2$

$y + z - 2 = 4$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

Step 3.2

Divide each term in $2 z = 2$ by $2$ and simplify.

Tap for more steps...

Step 3.2.1

Divide each term in $2 z = 2$ by $2$ .

$\frac{2 z}{2} = \frac{2}{2}$

$y + z - 2 = 4$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 z}{2} = \frac{2}{2}$

$y + z - 2 = 4$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

Step 3.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{2}{2}$

$y + z - 2 = 4$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

$z = \frac{2}{2}$

$y + z - 2 = 4$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

$z = \frac{2}{2}$

$y + z - 2 = 4$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Divide $2$ by $2$ .

$z = 1$

$y + z - 2 = 4$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

$z = 1$

$y + z - 2 = 4$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

$z = 1$

$y + z - 2 = 4$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

$z = 1$

$y + z - 2 = 4$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

Step 4

Replace all occurrences of $z$ with $1$ in each equation.

Tap for more steps...

Step 4.1

Replace all occurrences of $z$ in $y + z - 2 = 4$ with $1$ .

$y + 1 - 2 = 4$

$z = 1$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Simplify $y + 1 - 2$ .

Tap for more steps...

Step 4.2.1.1

Remove parentheses.

$y + 1 - 2 = 4$

$z = 1$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

Step 4.2.1.2

Subtract $2$ from $1$ .

$y - 1 = 4$

$z = 1$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

$y - 1 = 4$

$z = 1$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

$y - 1 = 4$

$z = 1$

$x + 3 y - 3 z + 8 = 3$

$w = 2$

Step 4.3

Replace all occurrences of $z$ in $x + 3 y - 3 z + 8 = 3$ with $1$ .

$x + 3 y - 3 \cdot 1 + 8 = 3$

$y - 1 = 4$

$z = 1$

$w = 2$

Step 4.4

Simplify the left side.

Tap for more steps...

Step 4.4.1

Simplify $x + 3 y - 3 \cdot 1 + 8$ .

Tap for more steps...

Step 4.4.1.1

Multiply $- 3$ by $1$ .

$x + 3 y - 3 + 8 = 3$

$y - 1 = 4$

$z = 1$

$w = 2$

Step 4.4.1.2

Add $- 3$ and $8$ .

$x + 3 y + 5 = 3$

$y - 1 = 4$

$z = 1$

$w = 2$

$x + 3 y + 5 = 3$

$y - 1 = 4$

$z = 1$

$w = 2$

$x + 3 y + 5 = 3$

$y - 1 = 4$

$z = 1$

$w = 2$

$x + 3 y + 5 = 3$

$y - 1 = 4$

$z = 1$

$w = 2$

Step 5

Move all terms not containing $y$ to the right side of the equation.

Tap for more steps...

Step 5.1

Add $1$ to both sides of the equation.

$y = 4 + 1$

$x + 3 y + 5 = 3$

$z = 1$

$w = 2$

Step 5.2

Add $4$ and $1$ .

$y = 5$

$x + 3 y + 5 = 3$

$z = 1$

$w = 2$

$y = 5$

$x + 3 y + 5 = 3$

$z = 1$

$w = 2$

Step 6

Replace all occurrences of $y$ with $5$ in each equation.

Tap for more steps...

Step 6.1

Replace all occurrences of $y$ in $x + 3 y + 5 = 3$ with $5$ .

$x + 3 (5) + 5 = 3$

$y = 5$

$z = 1$

$w = 2$

Step 6.2

Simplify the left side.

Tap for more steps...

Step 6.2.1

Simplify $x + 3 (5) + 5$ .

Tap for more steps...

Step 6.2.1.1

Multiply $3$ by $5$ .

$x + 15 + 5 = 3$

$y = 5$

$z = 1$

$w = 2$

Step 6.2.1.2

Add $15$ and $5$ .

$x + 20 = 3$

$y = 5$

$z = 1$

$w = 2$

$x + 20 = 3$

$y = 5$

$z = 1$

$w = 2$

$x + 20 = 3$

$y = 5$

$z = 1$

$w = 2$

$x + 20 = 3$

$y = 5$

$z = 1$

$w = 2$

Step 7

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 7.1

Subtract $20$ from both sides of the equation.

$x = 3 - 20$

$y = 5$

$z = 1$

$w = 2$

Step 7.2

Subtract $20$ from $3$ .

$x = - 17$

$y = 5$

$z = 1$

$w = 2$

$x = - 17$

$y = 5$

$z = 1$

$w = 2$

Step 8

Solve the system of equations.

$x = - 17 y = 5 z = 1 w = 2$

Step 9

List all of the solutions.

$x = - 17, y = 5, z = 1, w = 2$